Because of the importance of control techniques of three-phase electrical motors in a star configuration, reference will be made to a motor of this kind, however the same considerations apply, with the respective differences having been considered, for any poly-phase electrical load in any configuration.
One of the most widely used techniques for the control of three-phase electric motors is the FOC (Field Oriented Control). This technique, based on so-called SVM, induces sinusoidal voltages and currents in the windings of the motor and requires an accurate measurement of the rotor position and of the phase currents of the motor.
FIG. 1 depicts a block diagram of a system for controlling the motor torque by the FOC technique. When the windings of the motor are connected in star configuration, it is sufficient to measure only two phase currents for reconstructing the values of all three-phase currents.
The phase currents of the motor ia and ib may be measured at the same instant of every modulation period. The most commonly used current sensors are sensing resistors, Hall effect sensors and current transformers. Sensing resistors are preferred in most applications because they allow a sufficiently accurate sensing and are comparably the most economic choice.
Hereinafter reference will be made to current sensing resistors, though the addressed technical problems affect also any other kind of current sensor. Normally, the signal generated by a current sensor is sampled by one or more A/D (analog/digital) converters and the performance of the control system depends primarily on the precision with which these currents are measured.
When using an inverter for driving the windings of a motor, inevitably voltage harmonics superposed on the main voltage frequency are generated. These drive voltage harmonics produce current harmonics that disturb the measurement of the current in the windings. In order to prevent or reduce spurious effects due to current harmonics, the currents should be measured at certain instants. It has been shown, as in Richardson J., “Implementation of a PWM Regular Sampling Strategy for AC Drives,” that there is no contribution of harmonics superposed on the main component of the current at the beginning (instant 0) and at half (T/2) of the modulation period T. As an alternative, a method that contemplates repeated current measurements during the same period for estimating the values at instants 0 and T/2 as integral average values for the modulation period is known, as disclosed in Blasko et al., “Sampling Methods for Discontinuous Voltage and Current Signals and Their Influence on Bandwidth of Control Loops of Electrical Drives.”
The method disclosed in Blasko et al. can be used for estimating two currents flowing through two windings at a same instant using two A/D converters.
Using two A/D converters is an expensive approach and it may be desirable to use a single multiplexed A/D converter, i.e. alternately connected first to a winding then to the other winding through a common multiplexer.
However, the use of a single multiplexed A/D converter raises some problems. One of the two currents is inevitably measured at a different instant of the other current. Therefore the so measured values may not be sufficiently correct and may reduce the accuracy of control. The time lag with which the pair of measured current values are sensed depends on the conversion time of the A/D converter being used and the measurement error will contain a term that depends on the rate of variation of the current at the main frequency.
Moreover, even if the second measurement is carried out as soon as possible, it will be carried out at a later instant of the instant (0 or T/2) at which the harmonics are null, thus the measurement error will contain even an undesired contribution due to current harmonics superposed on the current at the main frequency.
The error due to the variation of the current at the main frequency can be estimated by assuming the drive current is sinusoidal. The error ΔI can be defined as follows:ΔI=I(t)−I(t0)wherein t0 is the instant in which the measurement was to be executed, andt=t0+Δt is the instant in which the measurement has been effectively carried out, wherein Δt is the conversion time of the A/D converter. Considering that:I(t)=IM·sin(ω·t+φ)whereinω=2πf=2π/T ΔI(t)=IM·(sin(ω·t+φ)−sin(ω·t0+φ))and that the initial phase φ can be fixed equal to 0,ΔI(t)=IM·(sin(ω·t)−sin(ω·(t−Δt)))the first time derivative of the error is
                    ⅆ        Δ            ⁢                          ⁢              I        ⁡                  (          t          )                            ⅆ      t        =            I      M        ·    ω    ·          (                        cos          ⁡                      (                          ω              ·              t                        )                          -                  cos          ⁡                      (                          ω              ·                              (                                  t                  -                                      Δ                    ⁢                                                                                  ⁢                    t                                                  )                                      )                              )      The maximum relative error is determined by imposing
                              ⅆ          Δ                ⁢                                  ⁢                  I          ⁡                      (            t            )                                      ⅆ        t              =    0    thus      t    =                            Δ          ⁢                                          ⁢          t                2            +      kT                          Δ        ⁢                                  ⁢                  I          MAX                            I        M              =                            Δ          ⁢                                          ⁢                                    I              MAX                        ⁡                          (                              Δ                ⁢                                                                  ⁢                                  t                  /                  2                                            )                                                I          M                    =              2        ·                  sin          ⁡                      (                                                            π                  ·                  Δ                                ⁢                                                                  ⁢                t                            T                        )                              
For a small size motor with four polar pairs, running at the speed of 10000 rpm, the current frequency isf=667 Hzthus if
                    Δ        ⁢                                  ⁢        t            T        =          10              -        3              then                    Δ        ⁢                                  ⁢                  I          MAX                            I        M              =          6.28      ·              10                  -          3                    Considering that the resolution of a 10 bit A/D converter is1/1023=9.78·10−4 and that the precision of a sensing resistor is typically 0.1%, the error is certainly non-negligible. Apparently, it is not possible to estimate in a simple way and with sufficient precision currents flowing in a same instant through two windings of a three-phase motor or of any poly-phase electrical load, using the same measuring instrument connected once with a winding, then with another winding.